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74 2. Elliptic curves Exercise 2.12.19. Let e1,e2,e3 be three distinct integers. Show that Δ = (e1 − e2)(e2 − e3)(e1 − e3) is always even. Exercise 2.12.20. In this exercise we study the structure of the quotient G/2G, where G is a finite abelian group. (1) Let p ≥ 2 be a prime and let G = Z/peZ, with e ≥ 1. Prove that G/2G is trivial if and only if p 2. (2) Prove that, if G = Z/2eZ and e ≥ 1, then G/2G ∼ Z/2Z. (3) Finally, let G be an arbitrary finite abelian group. We define G[2∞] to be the 2-primary component of G, i.e., G[2∞] = {g ∈ G : 2n · g = 0 for some n ≥ 1}. In other words, G[2∞] is the subgroup of G formed by those elements of G whose order is a power of 2. Prove that G[2∞] ∼ Z/2e1Z ⊕ Z/2e2Z ⊕ · · · ⊕ Z/2erZ for some r ≥ 0 and ei ≥ 1 (here r = 0 means G[2∞] is trivial). Also show that G/2G ∼ (Z/2Z)r. Exercise 2.12.21. Show that the space C : 2Y 2 − X2 = 34, Y 2 − Z2 = 34 does not have any rational solutions with X, Y, Z ∈ Q. (Hint: modify the system so there are no powers of 2 in any of the denominators, then work modulo 8.) Exercise 2.12.22. For the following elliptic curves, use the method of 2-descent (as in Proposition 2.10.3 and Example 2.10.4) to find the rank of E/Q and generators of E(Q)/2E(Q). Do not use Sage: (1) E : y2 = x3 − 14931x + 220590. (2) E : y2 = x3 − x2 − 6x. (3) E : y2 = x3 − 37636x. (4) E : y2 = x3 − 962x2 + 148417x. (Hint: use Theorem 2.7.4 first to find a bound on the rank.) Exercise 2.12.23. Find the rank and generators for the rational points on the elliptic curve y2 = x(x + 5)(x + 10).